Conjugacy class structure in the unitriangular matrix group

Note that the characteristic polynomial of all elements in this group is , hence we do not devote a column to the characteristic polynomial.

For reference, we consider matrices of the form:

Nature of conjugacy class

Jordan block size decomposition

Minimal polynomial

Size of conjugacy class

Number of such conjugacy classes

Total number of elements

Order of elements in each such conjugacy class

Type of matrix

identity element

1 + 1 + 1 + 1

1

1

1

1

non-identity element, but central (has Jordan blocks of size one and two respectively)

2 + 1

1

,

non-central, has Jordan blocks of size one and two respectively

2 + 1

, but not both and are zero

non-central, has Jordan block of size three

3

if odd4 if

both and are nonzero

Total (--)

--

--

--

--

--

Grouping by conjugacy class sizes

Conjugacy class size

Total number of conjugacy classes of this size

Total number of elements

Cumulative number of conjugacy classes

Cumulative number of elements

1

(total)

(total)

Conjugacy classes with respect to the general linear group

If we consider the action of the general linear group by conjugation, then there is considerable fusion of conjugacy classes. Specifically, there are only three equivalence classes, corresponding to the set of unordered integer partitions of 3 describing the possible Jordan block decompositions.